Movable space

A compact space , embedded in the Hilbert cube , is movable provided every neighbourhood of in admits a neighbourhood of such that, for any other neighbourhood of , there exists a homotopy with , . In other words, sufficiently small neighbourhoods of can be deformed arbitrarily close to [a2]. K. Borsuk proved that movability is a shape invariant. The solenoids (cf. Solenoid) are examples of non-movable continua.

The question whether movable continua are always pointed movable is still (1998) open.

For movable spaces various shape-theoretic results assume simpler form. E.g., if is a pointed shape morphism between pointed movable metric continua (cf. also Pointed space; Continuum; Shape theory), which induces isomorphisms of the shape groups , for all and if the spaces , are finite-dimensional, then is a pointed shape equivalence. This is a consequence of the shape-theoretic Whitehead theorem (cf. also Homotopy type; Homotopy group) and the fact that such an induces isomorphisms of homotopy pro-groups [a6], [a5].

Borsuk also introduced the notion of -movability. A compactum is -movable provided every neighbourhood of in admits a neighbourhood of in such that, for any neighbourhood of , any compactum of dimension and any mapping , there exists a mapping such that and are homotopic in . Clearly, if a compactum is -movable and , then is movable. Moreover, every -compactum is -movable [a3]. The notion of -movability was the beginning of the -shape theory, which was especially developed by A.Ch. Chigogidze [a4] (cf. also Shape theory). The -shape theory is an important tool in the theory of -dimensional Menger manifolds, developed by M. Bestvina [a1].

References

[a1]	M. Bestvina, "Characterizing -dimensional universal Menger compacta" Memoirs Amer. Math. Soc. , 71 : 380 (1988) pp. 1–110
[a2]	K. Borsuk, "On movable compacta" Fund. Math. , 66 (1969) pp. 137–146
[a3]	K. Borsuk, "On the -movability" Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. , 20 (1972) pp. 859–864
[a4]	A.Ch. Chigogidze, "Theory of -shape" Uspekhi Mat. Nauk , 44 : 5 (1989) pp. 117–140 (In Russian)
[a5]	J. Dydak, "The Whitehead and the Smale theorems in shape theory" Dissert. Math. , 156 (1979) pp. 1–55
[a6]	J.E. Keesling, "On the Whitehead theorem in shape theory" Fund. Math. , 92 (1976) pp. 247–253